A Hölder continuity result for a class of obstacle problems under non standard growth conditions

✍ Scribed by Michela Eleuteri; Jens Habermann

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 282 KB
• Volume: 285
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.201190020

✦ Synopsis

Abstract

Let Ω~1~ and Ω~2~ be bounded, connected open sets in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^N$\end{document} with continuous boundary, and let p > 2. We show that every positive linear isometry T from W^1, p^(Ω~1~) to W^1, p^(Ω~2~) that satisfies \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^{1,p}_0(\Omega _2) \subset TW^{1,p}_0(\Omega _1)$\end{document} corresponds to a rigid motion of the space, i.e., Tu = u○ξ for an isometry ξ of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^N$\end{document}, and more precisely ξ(Ω~2~) = Ω~1~. We also prove similar results for less regular domains, and we obtain partial results also for p = 2.